ALGEBRAIC THINKING AND APPLICATIONS

Transcription

1 Section ALGEBRAIC THINKING AND APPLICATIONS Objective : Simplify Algebraic Expressions Involving One or Two Variables Students have great difficulty recognizing the differences among linear, quadratic, and constant terms in algebraic form. Exponents seem insignificant to them. Viewing each type of term as an area helps students visualize the role each term plays in an expression. The following activities provide experience with such visualization in the combining of like terms. It is assumed that students have already mastered the four operations with integers. Activity Manipulative Stage Materials Packets of variable and unit tiles (described in step below) Worksheet a Legal-sized plain paper or light tagboard (for building mats) Regular paper and pencils Procedure. Give each pair of students a packet of tiles, two copies of Worksheet a, and a sheet of plain paper or tagboard (approximately 8.5 inches by 4 inches) for a building mat. If preferred, laminate the mats to make them more durable. Mats define a specific space on which to represent a problem being solved. If teacher-made tiles are used, each packet should contain the following in different colors of laminated tagboard: 8 square (quadratic) variable tiles, each inches by inches (color #); 8 square variable tiles, each.25 inches by.25 inches (color #2); 2 rectangular (linear) variable tiles, 0.75 inches by inches (color #); 2 rectangular variable tiles, 0.75 inch by.25 inches (color #2); and 20 unit tiles, 0.75 inch by 0.75 inch (color #). Each tile should have a large X drawn on one side to show the inverse of that tile. Use tagboard that is thick enough so that the X will not show through to the other side. Commercial tiles are also available for two different variables, but a large X must be drawn on one of the largest faces of each tile in order to represent the inverse of that tile when the X faces up. COPYRIGHTED MATERIAL

2 2 Math Essentials, High School Level 2. The meaning of a large square tile needs to be connected to a long rectangular tile of the same color. Have students place a rectangular variable tile of color # (call it variable A) horizontally on the mat. Then have them place two more variable tiles below and parallel to the first tile on the mat. Ask: If a single variable tile A is considered to cover an area of by A, or A, how can we describe the arrangement indicated by these tiles on the mat? ( rows of A. ) What product or area is this? ( A. ) Ask: How can we show A rows of A on the mat if we do not know what the value of A is? Show students how to build several rows of one variable tile each, using one variable tile A as the multiplier, or ruler, that indicates when to stop putting tiles in the product on the mat (see the illustration below). When the product is finished, the multiplier tile should be removed from the mat. Depending on the dimensions used to make the tiles, whether commercial or teacher-made, the width across several rectangular tiles placed with their longer sides touching may or may not match the length of the longer side of the same type of tile. Such a match is not important and should be deemphasized since the variable tile A is not considered to have a specific length or value in unit tiles. Therefore, although the width of 4 of the variable tile A may appear to match to one variable tile length as shown on the mat below, do not allow students to say that 4 rows of A equal 4A.. Ask: Is there another single block that will cover the same surface area on the mat that the product A of A, or A(A), covers? ( Yes. The large square tile in color #; its side length equals the length of the variable tile A. ) Again, discuss the idea that the large square tile in color # may or may not fit perfectly on top of the A rows of A tile arrangement; it will be close enough. Since both the square and rectangular tiles in color # are representing variables without known values, we want to maintain their variable nature as much as possible. Physical models like the tiles naturally have specific dimensions that affect or limit areas being built with the tiles, but for our purpose, we will assume that only the unit tiles may be used to represent exact amounts of area. We will now assign the large square tile in color # the name of A-squared, or A 2. Hence, A rows of A equal A 2. From now on, whenever A rows of A are needed, the large square tile will be used to show that amount of area on the mat. product product of variable A, or area A A of variable A, or area A(A) remove multiplier tile after product is built 4. Similarly the areas of the square and long rectangular variable tiles in color #2 might be described as B-squared, or B 2, and B, respectively. If an X appears on the top side of a variable tile, the inverse or opposite of the tile s area will be indicated. For

3 Algebraic Thinking and Applications example, a B-squared tile with X on top will be called the opposite or inverse of B-squared and written as ( B 2 ). Each small square tile in color # represents an area of by, or square unit of area. If a given set of unit tiles all have an X showing for example, 5 tiles with X then the tile value will be the negative of 5 square units of area and written as ( 5). Note that area itself is an absolute measure, neither positive nor negative. Area, however, can be assigned a direction of movement in real applications; hence, we can consider the opposite or negative of a given area. 5. After the area of each type of tile is identified, have students do the exercises on Worksheet a. For each exercise, they should place a set of tiles on the building mat to show the first expression. Then they will either add more tiles to this initial set or remove some tiles from the set according to the second expression of the exercise. 6. After combining tiles that have the same amount of area, students should record an expression for the total or remaining area on the worksheet. 7. Discuss an addition exercise and a subtraction exercise with the class before allowing students to work the other exercises independently. Consider Exercise on Worksheet a for addition: _ A 2 + 2A- 5i+ _- A 2 + A+ 2i: Use color # variable tiles with the color # unit tiles. Have students place large square (quadratic) variable tiles, 2 long rectangular (linear) variable tiles, and 5 negative unit tiles on the building mat to represent the first expression. Any such group of tiles is called a polynomial, that is, a combination of variable tiles and/or unit tiles. Leaving this set of tiles on the mat, have students place additional tiles on the mat below the initial tiles to represent the second expression. The second set should contain a quadratic variable tile with X showing, linear variable tiles, and 2 unit tiles: Ask: Can any 0-pairs be made through joining, then removed from the mat? (One 0-pair of the large quadratic tiles and two 0-pairs of the small unit tiles should be formed and removed from the building mat.) Can you now describe the total in tiles still on the mat? Since tiles for two of A-squared, 5 of A, and remain on the mat, students should complete the recording of Exercise on Worksheet a: _ A 2 + 2A- 5i+ _- A 2 + A+ 2i = 2A 2 + 5A-. Now consider Exercise 2 on Worksheet a for subtraction: _ 4A 2 - A + 4i - _ A 2 + 2A - 2i. Again, use color # variable tiles with the unit tiles. Have students place tiles on their mats to show the first group. There should be 4 of the quadratic variable tile, of the linear variable tile with the X-side showing for the inverse variable, and 4 positive unit tiles on the building mat. Discuss the idea that the subtraction symbol between the two polynomial groups means to remove each term in the second group from the first group. Ask: Can we remove one quadratic variable tile from the original four? ( Yes; quadratic variable tiles, or A 2, will remain. ) Can 2A be removed

4 4 Math Essentials, High School Level from A? Since only inverse variable tiles are present initially, 0-pairs of A and A tiles will need to be added to the mat until two of the variable A are seen. Then 2A can be removed, leaving 5 of A on the mat. Similarly, 2 will be removed from +4 by first adding two 0-pairs of + and to the mat. Then 2 can be removed from the mat, leaving +6. The mat arrangement of the initial tiles and the extra 0-pairs of tiles is shown here before any tile removal occurs. Have students complete Exercise 2 on Worksheet a by writing an expression for the tiles left on the mat: _ 4A 2 - A+ 4i- _ A 2 + 2A- 2i = A 2-5A+ 6. Remind students that when they use 0-pairs of a tile and remove one form of the tile (for example, positive), then the other form (for example, negative) remains to be added to the other tiles on the mat. Show students that when they needed to remove 2A from the mat earlier, two 0-pairs of A and A were placed on the mat. After 2A was removed to show subtraction, the two inverse variable tiles, 2A, still remained on the mat to be combined with the other tiles for the final answer. Hence, a removal of a tile from the mat is equivalent to adding the inverse or opposite of that tile to the mat. To confirm this, have students place the original group of tiles _ 4A 2 - A+ 4i on the mat again. The opposites needed ( A 2, 2A, and +2) should then be placed on the mat and combined with the original tiles. See the illustration below. Remove any 0-pairs formed, leaving tiles for A 2, 5A, and +6 on the mat as the answer. Finally, have students write another equation below Exercise 2 on Worksheet a, this time showing the alternate method that uses addition: _ 4A 2 - A+ 4i+ _-A 2-2A+ 2i = A 2-5A+ 6. Encourage students to use whichever of these two methods seems comfortable to them. In the answer key for Worksheet a, when the coefficient of a final variable is, the number will be written with the variable. This approach seems to be helpful to many students. Nevertheless, discuss the idea with the class that the in such cases is often not recorded but simply understood as being there.

5 Algebraic Thinking and Applications 5 Answer Key for Worksheet a. 2A 2 + 5A- 2. A 2-5A+ 6; alternate: _ 4A 2 - A+ 4i+ _-A 2-2A+ 2i= A 2-5A+ 6. B ; alternate: _ 5B + i+ _-2B - i= B A 2 + A-4 5. B 2 + 2A A+ A + 5; alternate: A+ _ A + 5i= A+ A A 2 ; alternate: _ 5B - 4i+ _- 5B + 4+ A i= A 8. A+ A B

6 6 Worksheet a Building Sums and Differences with Tiles Name Date Build each polynomial exercise with tiles. Different variables require different tiles. Record the result beside the exercise. For each subtraction exercise, also write the alternate addition equation below the subtraction equation.. _ A 2 + 2A- 5i+ _- A 2 + A+ 2i= 2. _ 4A 2 - A+ 4i- _ A 2 + 2A- 2i=. 2 2 _ 5B + i- _ 2B + i= 4. _ 2A 2 - A+ i + ^4A- 5h = 5. _ 4A- 2+ B i + ^-6-2Ah = Copyright 2005 by John Wiley & Sons, Inc A-_ -A - 5i= _ 5B -4i-_ 5B -4- A i= A-2A - 5+ B- 8+ A =

7 Algebraic Thinking and Applications 7 Materials Worksheet b Regular paper and pencil Activity 2 Pictorial Stage Procedure. Give each student a copy of Worksheet b. Have students work in pairs, but they should draw the diagrams separately on their own worksheets. Large squares will be drawn for the quadratic variable, a long rectangle whose length equals an edge length of the large square will be drawn for the linear variable, and a small square will represent the integral unit. A large X should be drawn in the interior of a shape to show the inverse of that shape. If an exercise involves two different variables, letters need to be written on the drawn shapes to identify the different variables. The product of two different variables, for example, A and B, should be shown as a large rectangle similar in size to the quadratic squares and labeled as AB. The notation AB simply means A rows of B, or the area AB. 2. For addition exercises, students should draw the required shapes and connect any two shapes that represent a 0-pair. The remaining shapes will be recorded in symbols to show the sum.. For subtraction exercises, students will be asked to use either the removal method or the alternate method, which involves addition of inverses. To remove a shape, students should mark out the shape. When needed, two shapes should be drawn together as a 0-pair. For the alternate method, inverses of the subtrahend expression should be drawn and combined with the first expression to produce a sum. The result will be recorded symbolically. 4. When checking students work after all are finished, allow time for students to explain their steps; do not just check for answers. Students need to practice expressing their ideas mathematically. Such verbal sharing is also very beneficial to auditory learners. 5. Discuss Exercises and 2 on Worksheet b with the class before allowing partners to work together on their own. Consider Exercise : _- B 2 + B+ 2i+ _ B 2-4B+ i. Students should draw the necessary shapes on their papers to represent each polynomial group. The shapes for the first polynomial group may be drawn in a row from left to right following the order of the given terms. The shapes for the second polynomial group should be drawn as a second row below the first row, but students may rearrange the shapes and draw them below other like shapes in the first row. Since only one variable is involved, no labeling is needed for the shapes. Any 0-pairs should be connected. Remaining shapes will then be counted and recorded as the answer. A sample drawing is shown here:

8 8 Math Essentials, High School Level The final equation will be as follows and should be recorded on Worksheet b: _- B 2 + B+ 2i+ _ B 2-4B+ i = -2B 2 - B+. At this point, begin to encourage students to record the terms of a polynomial with their exponents in decreasing order. Now consider Exercise 2: _ A 2 + 5A-i- _ 2A 2 + A+ 2i. Since the removal process is required for this exercise, students should draw shapes for the first polynomial group and then draw any 0-pairs below that group, which will be needed in order to mark out the shapes shown in the second group. The shapes remaining or not marked out in the finished diagram will be the difference. Here is the completed diagram: The final equation should be recorded on Worksheet b as follows: _ A 2 + 5A-i- _ 2A 2 + A+ 2i = - A 2 + 2A-5. It may be helpful for some students to write A 2 instead of A 2. This is acceptable notation. Answer Key for Worksheet b Only symbolic answers may be given B - B+ (see sample diagram in text) A + 2A- 5 (see sample diagram in text). 5A 2-2A A- 2B A + B -2 Sample diagram for Exercise 5: B 2 B 2 B 2 B 2 AB B 2 A 2 A 2 AB 6. B 2 -AB- A+

9 9 Worksheet b Drawing Sums and Differences Name Date Use shapes to simplify each polynomial exercise according to the directions. Label shapes to identify different variables when necessary. Record the algebraic result beside the exercise.. _- B 2 + B+ 2i+ _ B 2-4B+ i= 2. _ A 2 + 5A-i- _ 2A 2 + A+ 2i= Copyright 2005 by John Wiley & Sons, Inc. [use removal] 2 2. _ 6A - A+ i- _ A + A- i= [use addition of inverses] 4. ^A- 5B+ 6h+ ^B-4A- 5h= 5. _ 4B 2 + AB- 2i+ _ 2A 2 -B 2 - ABi= 6. _ 2B 2 - AB+ Ai- _ B 2 + 2A- i= [use either method]

10 0 Math Essentials, High School Level Materials Worksheet c Regular paper and pencil Activity Independent Practice Procedure Give each student a copy of Worksheet c. After all students have completed the worksheet, ask various students to show their solutions or any illustrations they might have used to the entire class. In particular, select students to share their work who have solved the same problem in different ways. Answer Key for Worksheet c. C 2. D. A 4. A 5. C Possible Testing Errors That May Occur for This Objective When combining polynomials, students fail to recognize 0-pairs among the terms; for example, they write the sum (+x) + ( x) as (+6x) instead of 0. When finding differences by the alternate method of adding inverses of the subtrahend group to the original minuend group, students do not exchange all the terms for their inverse forms; hence, they add the wrong terms together. For example, in (2N + 5) (N + K ), they actually add (2N + 5) to ( N + K + ) instead of to ( N K + ). Students make computational errors when combining like terms. For example, (+4y) + ( 7y) is incorrectly written as (+y) instead of ( y).

11 Worksheet c Finding Sums and Differences of Polynomials Name Date Solve the exercises provided and be ready to discuss your methods and answers with the entire class.. The following diagram represents the product of 2N rows of (N + 2). Which expression is equivalent to the total area, 2N(N + 2), of the product diagram? ruler 2N product diagram Copyright 2005 by John Wiley & Sons, Inc. A. 6N B. 6N + 2 C. 6N 2 + 4N D. 6N Which expression is equivalent to b x y y x 2 l_ 6 - i+ _ 4-7 i? A. - x+ y B. - x+ y C. 4x- 2y D. - x+ 2y. Which expression is equivalent to ^5k-2h^kh-^5k-2h^k-h? A. 0k 2 + k-2 B. 0k 2-4k-2 C. 0k 2-2 D. 5k Simplify the expression and evaluate for T = 5: _ 2T 2 + Ti+ _ 5T- 2T 2 + i =? A. 7 B. 8 C. 4 D. Not here Which expression is equivalent to _ 5y -xy-4i-_ 5y -y-xy-4i? A. 0 B. 6xy 4 C. y D. y

12 2 Math Essentials, High School Level Objective 2: Solve a Linear Equation Involving One Variable with a Fractional Coefficient Fraction operations are difficult for most students to comprehend. Extending fraction multiplication to partial sets of a variable is even more complicated. Rote methods are often taught and students seemingly master them, yet they do not attain a deeper understanding of the method. Students need experiences with partial sets that will lead to discovering what the whole set will contain. The following activities provide such experiences. Activity Manipulative Stage Materials Tile sets (minimal set: variable tile with its inverse tile, 0 positive unit tiles with their inverse tiles) Building Mat 2a Pieces of colored yarn (approximately 2 inches long) or flat coffee stirrers Extra construction paper (use colors that match the variable tiles in the sets) Scissors Regular paper and pencils Procedure. Give each pair of students a set of tiles, a copy of Building Mat 2a, a piece of yarn or coffee stirrer, scissors, and a sheet of construction paper (the same color as their variable tiles). 2. To make fractional variable tiles, have students cut out 6 rectangular strips from their sheet of construction paper. The paper strips should be the same size and color as one of their variable tiles. Show them how to fold the paper strips, mark the creases, and label the parts with fractional names. Two strips should be folded and labeled for halves, yielding 4 half-variables total. Two more strips should be folded and labeled for thirds, and another 2 strips for fourths. Use the ratio format for labeling the fractional parts, for example, 2,, and 4. On one side of each fractional part made, have students mark a thin, large X to represent the inverse of the fractional part. Be sure that the X does not show through on the other side of the paper. Additional paper strips may be cut out as needed.. Have students place tiles on Building Mat 2a to build each equation shown on Worksheet 2a. Below each equation on Worksheet 2a, they should record the symbolic steps they used to solve the equation. 4. For each equation solved, students should confirm their solution. Have them rebuild the original equation with tiles and then substitute the variable s discovered value in unit tiles for the variable tile itself to show the true equality. Students should write a check mark beside the solution on the worksheet after the correct value has been confirmed. 5. Discuss Exercises and 2 on Worksheet 2a with the class before allowing students to work the rest of the exercises independently. Consider Exercise on Worksheet 2a: K 2 - ^+ h = -7. Have students build this equation with tiles. A paper variable folded into thirds should be cut apart and 2 of the thirds placed on the left side of Building Mat 2a. The subtraction sign on the left side of the equation indicates that + must be removed from the mat. Since there are not positive unit tiles on the left side, three 0-pairs of positive and negative unit tiles should be placed on

13 Algebraic Thinking and Applications the left side of the mat, followed by removal of the positive unit tiles. Some students may realize that subtracting + is equivalent to adding ; if so, they may simply place on the left side of the mat with the fractional variable tiles instead of working with the 0-pairs first. Finally, 7 negative unit tiles should be placed on the right side of the mat. Ask: What changes can we make to the mat that will leave the variable tiles by themselves on the left? ( Either take away or bring in + on the left. ) The students must then repeat whichever action they choose on the right side of the mat and record their steps with symbolic notation below Exercise on Worksheet 2a. (a) Take-Away Model (b) Add-On Model K 2 - ^+ h= -7 K 2 - ^+ h= -7 K 2 + ^- h= -7 K 2 + ^- h= ^ h -- ^ h + ^+ h + ^+ h K 2 =-4 K 2 =-4 remove - from both sides of mat add + to both sides of mat Ask: Two of the thirds of a variable remain on the left by themselves. What do we need to do to find a whole variable tile now? ( First find a single third of the variable by separating the 2 variable parts or by finding half of the present variable group; of this single third will make a whole variable tile. ) Remember that you need to use multiplication language here, not addition language. That is, you want of the third-variable in order to make a whole variable, not 2 more of the third-variable. triple

14 4 Math Essentials, High School Level After students perform these two actions, halving and then tripling, on both sides of the mat, have them record their new actions as shown below: (a) Take-Away Model (b) Add-On Model K 2 - ^+ h= -7 K 2 - ^+ h= -7 K 2 + ^- h= -7 K 2 + ^- h= ^ h -- ^ h + ^+ h + ^+ h K 2 =-4 K 2 =-4 K 2 2 b = -4 l ^ h K b = -4 l ^ h 2 K =-2 K =-2 b K = -2 l ^ h b K = -2 l ^ h K =-6 K =-6 Note: The hope is that most students will realize that the halving-tripling process used above can be combined and shown in the recording as follows: K 2 =-4 K 2 2 b = -4 l ^ h 2 K =-6 If the students do not easily accept this combination of steps, allow them to continue recording the two separate steps of division and multiplication. Also notice that when we previously needed half of the 2-thirds of the variable K, the factor was 2 used with multiplication, rather than using 2 as a divisor, or dividing by 2. This helps students connect to the reciprocal method more easily. The solution, K = 6, should be confirmed on the mat by exchanging variable parts for units. Have students rebuild the original equation on their mats and place 6 negative unit tiles on the mat just above the variable tiles. Since thirds are involved, 6 should be separated into equal groups of 2 each. Each variable part on the mat should be replaced with one of the groups of 2. The unused 2 should be removed from the top of the mat. Now 2 groups of 2, along with another group of, can be seen on the left side, and 7 appears on the right. Since the two sides have the same total value, K = 6 is the correct solution. A check mark should be written beside the solution equation on the worksheet. Now discuss the equation for Exercise 2 on Worksheet 2a: ^- h - p = Have students build the equation on Building Mat 2a, using paper fourths of a variable. Each fourth-variable should have an X marked on one side. First have students

15 Algebraic Thinking and Applications 5 place 4 in unit tiles on the right side of the mat and in unit tiles on the left side. Discuss the idea that just as with unit tiles earlier, subtraction of a variable group is equivalent to the addition of the inverse variable group. Hence, after showing a 0-pair of fourths of a variable (one plain fourth and one fourth with an X-side facing up) on the left side of the mat with and then removing + p, students will have - p and 4 4 still remaining on the left side. 4 Students should now isolate the fourth of an inverse variable by either the takeaway or the add-on method. The tiles on each side of the mat should then be quadrupled to yield 4 variable parts that are equivalent to a whole inverse variable, p. If students prefer, they may replace the 4 variable parts on the mat with a whole inverse variable tile. Since a solution usually involves a value for the regular variable p, not its inverse, p, a coffee stirrer or piece of yarn should be placed below the last row of tiles on the mat: p = 2. The inverse of p is p and the inverse of 2 is +2; therefore, the regular, whole variable tile should be placed on the mat below the coffee stirrer or yarn and the 4 fourths of the inverse variable, and 2 positive units should be placed below the negative units and coffee stirrer or yarn on the right. The building and recording are as follows: (a) Add-On Method (b) Take-Away Method ^- h- p = -4 4 ^- h- p = -4 4 ^- h + b- p = -4 4 l ^- h + b- p = -4 4 l + ^+h + ^+ h -- ^ h -- ^ h - p = p = b- p = 4^- 4 l h 4 b- p = 4^- 4 l h - p = -2 - p = -2 So, p =+ 2 So, p = remove 0-pairs remove - from both sides of mat

16 6 Math Essentials, High School Level Here are the final steps for both methods: yarn or coffee stirrer Finally, the solution needs to be confirmed. The original equation should be rebuilt on the mat, including one of the fourths of the inverse variable tile. If p = +2, then p must equal 2; hence, 2 should be placed on the mat above the tiles on the left side. Since a fourth of the variable is needed, 2 should be separated into four equal groups of each. One group of should be exchanged for the variable part and the other three groups of removed from the mat. A single, negative unit tile, as well as the group of, are now seen on the left side of the mat, and a group of 4 is seen on the right. Both sides of the mat have the same total value, so the solution, p = +2, is correct. A check mark should be written beside the solution equation on the worksheet. Answer Key for Worksheet 2a Only solutions are provided; no mats are shown.. K = 6 2. p = +2. C = m = 4 5. B = +4 [use half-variable tiles; then isolate the variable tiles on the left side of the mat; separate each side of the mat into equal groups to find the value for one half-variable tile] 6. w = 2

17 7 Building Mat 2a Copyright 2005 by John Wiley & Sons, Inc.

18 8 Worksheet 2a Solving Linear Equations with Tiles Name Date Solve the equations with tiles on a building mat. Below each equation, record the steps used with symbolic notation. Confirm each solution found by exchanging the appropriate amount of unit tiles for the variable tiles given in the original equation. Write a check mark beside any solution equation shown to be correct with tiles.. K 2 - ^+ h= = - m 4 -^-h 2. ^- h- p = B + 6= Copyright 2005 by John Wiley & Sons, Inc.. 7= 2+ C 6. ^- 5h + w 2 b- 4 2 l = +

19 Algebraic Thinking and Applications 9 Materials Worksheet 2b Regular pencils Red pencils Activity 2 Pictorial Stage Procedure. Give each student a copy of Worksheet 2b and a red pencil. Have students work in pairs. 2. Have students draw a small square to represent each unit integer. For a negative unit, they should draw diagonals inside the square. Tall, narrow rectangles will represent whole variable bars. To show a fractional amount of a variable, have students draw a short rectangle (but slightly taller than the squares drawn for integral units) and write the fractional label inside the rectangle. A large but light X should be drawn inside the rectangle to show the inverse form of the variable.. For each equation on Worksheet 2b, students should draw a diagram for the equation on the worksheet. They should transform the diagram in order to find the solution to the equation. 4. After each new step has been performed on the diagram, have students record that step in symbols beside the diagram on Worksheet 2b. 5. After a solution is found, students should confirm the solution by drawing in red pencil the appropriate number of small squares on each variable shape in the diagram. The total number of small squares on the left side of the frame should equal the total number of small squares on the right side. Also have them confirm their solutions by writing a number sentence below the symbolic steps to show the substitutions used. 6. Discuss Exercise on Worksheet 2b with the class before allowing students to work independently. Consider the equation for Exercise : - M + 5= + 7. To make a diagram of this equation, ask students to draw a short rectangle (but slightly taller than the squares drawn for integral units) on the left side of a pair of parallel line segments (the equal sign). Have them draw a light but large X (to show the inverse) in the interior of the rectangle and write the fraction,, over the X. This will represent - M, read as one-third of the inverse of the variable M. Avoid the language negative one-third M until later. Also draw 5 small, plain squares on the left side with the variable rectangle. Seven small, plain squares should then be drawn on the right side to represent +7. The diagram should be drawn on the left half of Worksheet 2b below the equation.

20 20 Math Essentials, High School Level To isolate the variable by itself, students have the usual two methods available: remove +5 from both sides of the diagram, or bring in 5 to both sides of the diagram to form 0-pairs of the units. The transformed diagrams will appear as follows: (a) Removal Method (b) Add-On Method A horizontal bar should be drawn below the initial diagram that shows the removal or add-on step, and the shapes remaining for - M on the left and +2 on the right should be redrawn below that bar. Students now need to form a whole variable. This is done by drawing more rows of shapes on the second diagram until there are rows in all; each row shows a - M on the left of the vertical bars and +2 on the right. Remind students that they now have times as many - M s and times as many groups of +2 as they did before they drew the extra amounts on the diagram. times as many is multiplicative language, which is needed for this type of equation. By this time, students should recognize the three inverse thirds of a variable, one drawn above another, as a whole variable, and they should not need to draw a new, longer rectangle to show the whole inverse variable. It might be helpful, however, to have students draw a larger rectangle around the three smaller rectangles to show them grouped together. Since the diagram shows M = +6, a horizontal bar should be drawn below this diagram and a plain rectangle for M drawn below the bar on the left side and rows of 2 drawn below the bar on the right. That is, inverses have to be taken of M and +6 in order to solve for the regular variable, M.

21 Algebraic Thinking and Applications 2 Students should record their pictorial steps in symbolic notation on the right half of the worksheet below the equation. Depending on the method used, their recordings should appear as follows: (a) Removal-Multiplication Method (b) Add-On/Multiplication Method - M + ^+ 5h=+ 7 - M =+ 7 ^ h -+5 ^ h -+ ^ 5h +- ^ 5h +- ^ 5h - M = M = +2 b- M = ^+ 2 l h b- M = ^+ 2 l h - M = M = + 6 So, M =-6 So, M =-6 To confirm the solution as M = 6, have students first draw in red pencil two plain unit squares above the small rectangle for - M in the initial diagram; that is, if M = 6, the fractional amount must equal +2. The left side of the initial diagram now contains +7 in small plain squares and the right side contains +7, which confirms the solution. Below the symbolic steps, students should write the number sentence that shows their substitution: - M + 5 ^+ h= ^+ 6h+ ^+ 5h= ^+ 2h+ ^+ 5h = + 7 [viewing - M as a third of the inverse variable, M, or of +6]; or - M ^+ h= - ^- h+ ^+ h= ^+ h+ ^+ h =+ [viewing - M as the opposite of of the variable, M, or - of 6]. The +7, found after substitution to be the value of the left side of the original equation, agrees with the +7 given for the right side of the equation. Thus, M = 6 is confirmed again to be the correct solution. Answer Key for Worksheet 2b Only solutions and possible substitution number sentences are provided here. No diagrams are shown except for a partial diagram from Exercise 2.. M = 6; - M ^+ h= ^+ h+ ^+ h= ^+ h+ ^+ h= + [The complete diagram and recordings are shown in the text.] 2. A = +2; A = 4 2 ^+ 2h- 8 = ^+ 6h- 8 = -2 [When multiple fractional parts of a variable are present in an equation and the variable parts are isolated by removing appropriate unit squares, rings should be drawn around the remaining shapes to form the same number of equal groups on each side of the diagram. Each fractional variable represents one group, so in this exercise, there will be two groups on the right side; this requires two groups of + to be formed on the left side. One group from each side is then redrawn and will be repeated to form the equivalent of a whole variable amount. Part of the diagram is shown here.]

22 22 Math Essentials, High School Level c = +8; c - 4 = ^+ 8h- 4= ^+ 2h- 4= A = 0; ^ + 5h b - A 5 l = ^ + 5h 5 ^ + 0h ^ + 5h ^ + 6h

23 2 Worksheet 2b Solving Linear Equations by Drawing Diagrams Name Date Solve each equation by drawing a diagram below the equation. Beside each diagram, record the steps used with symbolic notation. Confirm each solution found by drawing in red pencil the appropriate number of unit squares for each fractional variable above that variable shape in the initial diagram. Also, below the symbolic steps, write a number sentence that shows the substitutions made in the original equation.. - M = + 7 ^ h. - 2 = c -4 4 Copyright 2005 by John Wiley & Sons, Inc = A ^+ 5h + b- A 5 l = +

24 24 Math Essentials, High School Level Materials Worksheet 2c Regular pencils Activity Independent Practice Procedure Give each student a copy of Worksheet 2c to complete independently. After all have finished, have them share their methods and answers with the class. Answer Key for Worksheet 2c. C 2. A. n = X = c = 0 Possible Testing Errors That May Occur for This Objective Students do not correctly interpret the subtraction sign in an equation and use addition instead; for example, for the expression N -^-8h they will use N + ^-8h. Other sign errors also occur. When an equation involves a fractional coefficient with the variable, students attempt to use the reciprocal method, but do not multiply by the correct fraction. For example, when trying to solve K 4 =+ 6, they will use either K b 4 l or 4 b K 4 l instead of the combination step, 4 K b 4 l. When applying the reciprocal method to solve an equation, students will multiply the variable group by the reciprocal of the variable s coefficient, but fail to multiply the equivalent constant by the same value. As an example, for A 5 2 =-4 students will compute 5 2 A 2 b 5 l, but will continue to use 4 instead of ^ h.

25 25 Worksheet 2c Solving Linear Equations Name Date Complete each exercise provided. Be ready to explain to other students the steps or reasoning you used to work each exercise.. If A = +22 is a solution for the equation - A + 9= - 2, which expression may be 2 used to confirm the solution? A. - ^+ h+ 9 C ^ h B. - ^- 22h+ 9 D ^ h Copyright 2005 by John Wiley & Sons, Inc. 2. Which expression is not a correct interpretation of the expression - p 2? A. - b p 2 l B. -2 b p l C. p 2 _ - i D. 2 b- p l. Solve for n: + 7 = n -^-2 h. 4. Solve for X: - 8 = ^-6h - X Solve for c: 8 + c 6 2 = - 2.

26 26 Math Essentials, High School Level Objective : State an Equation with One or More Variables That Represent a Linear Relationship in a Given Situation; Apply the Equation to Solve the Problem, If Appropriate Practical applications of mathematics require that students be able to translate the actions of a situation into an equation. Students may know how to solve a linear equation written in symbolic language, but this in no way guarantees that they can translate a word problem into an equation. This lesson focuses on the translation, but some actual solving may be required for additional practice. It is assumed that students already have a basic knowledge of how to solve simple linear equations involving integers. Objective 2 provides specific training with fractional coefficients if a review is necessary. Activity Manipulative Stage Materials Sets of tiles (minimum set: 8 linear variable tiles of equal length, 0 unit tiles, and the fractional variable tiles from Objective 2; inverse tiles should be included for each type of tile) Building Mat a Worksheet a Regular pencils Procedure. Give each pair of students a set of tiles, a copy of Building Mat a, and two copies of Worksheet a. 2. Using the tiles, students should build equations on the building mat to represent the situations described in each exercise on Worksheet a.. For each exercise modeled with tiles, have students write the initial equations in symbols below the word problem. Then have them solve the equations with the tiles. The solutions should also be recorded beside the equations on Worksheet a. 4. Discuss Exercises and 2 on Worksheet a with the class before allowing students to work independently with their partners. Consider Exercise on Worksheet a: Three consecutive positive integers have a sum of 24. Find the three consecutive integers. Discuss examples of consecutive positive integers like 2,, and 4, or 25, 26, and 27. Guide students to recognize that each new number is one more than the previous number. Since the actual numbers are not yet known, have students place one variable tile on the left side of Building Mat a to represent the first number. The second number is one more than the first number, so students should place another variable tile (same color or tile length as the first variable tile), along with unit tile, on the left side of the mat. Finally, they should place an additional variable tile and 2 unit tiles on the left side to represent the third number. The collective set of tiles on the left is the sum of the numbers. Since the sum equals 24, students should place 24 unit tiles on the right side of Building Mat a. Here is the initial appearance of the building mat:

27 Algebraic Thinking and Applications 27 Have students record the following unsimplified equation below Exercise, using N for the variable tile: N+ ^N+ h+ ^N+ 2h = 24. Students should then solve the equation with the tiles. Remind them that the variable tiles need to be isolated, which requires that the unit tiles be removed from the left side of the mat. To keep the building mat balanced, unit tiles must also be removed from the right side. Continuing, the remaining variable tiles must be separated into groups (each variable tile forms a group ), which forces the 2 unit tiles on the right side to be separated into groups as well. The groups on the mat should now have this appearance: Each variable tile determines a row on the right side of the mat. Hence, the value of one variable tile equals +7, which becomes the first number of the three consecutive numbers being sought. The next two numbers are represented by N + and N + 2, so their values are 7 + = 8 and = 9. Students should record the following three equations beside their initial equation for Exercise on Worksheet a: N = 7, N + = 7 + = 8, and N + 2 = = 9. Now consider Exercise 2 on Worksheet a: Three-fourths of the Math I class and five students from the Math II class are planning to go to the museum. Seventeen students in all will go on the field trip. How many students total attend the Math I class? Have students use their paper strips from Objective 2 to show fractional amounts of a variable tile. The whole variable tile equals the number of students in Math I, so of the fourth-variable tiles should be placed on the left side of Building Mat a to represent the students from Math I going on the field trip. Five unit tiles should also be placed on the left side to show the 5 students from Math II who will be going. Since the combined groups equal 7, students should place 7 unit tiles on the right side of the building mat. The building mat will have the following initial appearance: Using the variable M as the total number of students in Math I, have students record the following equation below Exercise 2 to represent the tiles shown on the building mat: M 4 + 5= 7.

28 28 Math Essentials, High School Level Now have students use the tiles on the mat to solve for the value of the whole variable, M. At first they must remove 5 unit tiles from both sides of the mat to isolate the fourth-variable group on the left side. Then the fourth-variables should be separated into three equal groups (a fourth-variable forms a group this time), which forces the 2 remaining unit tiles on the right side to be separated into three equal groups as well. The new tile arrangement will be as shown: Discuss the idea that one row on the building mat shows that a fourth-variable tile equals +4. To make a whole variable tile, four of the fourth-variable tiles will be needed on the left side of the mat. Similarly, four rows of +4 will be needed on the right side to keep the mat balanced. Have students place more tiles on the building mat in order to have the four rows needed. The solution to the initial equation is now found. Four fourthvariable tiles or whole variable tile equals 4 rows of +4 each. Have students record the following statement beside the original equation below Exercise 2: M = 6 students in Math I. Answer Key for Worksheet a. N+ ^N+ h+ ^N+ 2h = 24; N= 7, N+ = 7+ = 8, N+ 2= 7+ 2= 9 2. M 4 + 5= 7; M = 6 students in Math I. 4p - 6= 0; p = 4 cents per piece of gum 4. R - = 4; R = 4 rings in a full box 2 5. T+ ^T+ 4h+ T= 9; T= movie passes for Toni 6. G= 8; G= 24 gallons in a full drum

29 29 Building Mat a Copyright 2005 by John Wiley & Sons, Inc.

30 0 Worksheet a Modeling Linear Relationships with Tiles Name Date For each exercise, build an equation with tiles on Building Mat a to represent the situation. Write the equation in symbols below the exercise. Solve the equation with tiles, and write the solution beside the symbolic equation.. Three consecutive positive integers have a sum of 24. Find the consecutive integers. 2. Three-fourths of the Math I class and 5 students from the Math II class are planning to go to the museum. Seventeen students in all will go on the field trip. How many students are in the Math I class?. Kate bought 4 pieces of chewing gum at the school store. After a discount of 6 cents was applied to the total purchase, she paid 0 cents in all. What was the original price for each piece of gum? 4. Jorge won one-half of a box of rings at the carnival, but on the way home lost of the rings. Later at home, he still had 4 rings left. How many rings were originally in a full box? Copyright 2005 by John Wiley & Sons, Inc. 5. Maria has 4 more movie passes than Toni. Angela has times as many passes as Toni. Together the three girls have a total of 9 movie passes. How many movie passes does Toni have? 6. Two-thirds of the oil in an oil drum has leaked out. Eight gallons of oil are left in the drum. How many gallons total can the drum hold?

31 Algebraic Thinking and Applications Materials Worksheet b Regular pencils Regular paper Activity 2 Pictorial Stage Procedure. Give each student a copy of Worksheet b. Have students work in pairs. 2. For each exercise on Worksheet b, have students draw on regular paper a diagram for each relationship described in the given situation. If more than one type of variable is needed, students will need to label each variable shape with a letter they select for that variable.. Beside each diagram, students should record an equation in symbolic language that is equivalent to the diagram. 4. For practice in solving equations, students might also be instructed to solve for the variables in the equations they find for Exercises 2 through 5 and Exercise Discuss Exercises and 2 on Worksheet b before allowing students to work independently with a partner. Consider Exercise on Worksheet b: Eddie s Dogwalking Service charges $ for each walk plus $2 per hour for each hour the dog is walked. Find an equation that shows the relationship between the number of hours walked, H, and the total cost, C, for one walk. Discuss the idea that the total cost is the combination of the single fee of $ and the charges based on time. Each hour is worth $2, so H hours indicates how many of the $2 are needed. That is, the H hours serves as the multiplier, or counter of sets, and $2 is the multiplicand, or the set being repeated. Have students draw an equation frame that shows a variable C on the left side and shapes for the sum of the two kinds of charges on the right side. A pair of vertical parallel bars should be drawn to indicate equality of the two sides of the diagram. Because the multiplier in this case is a variable or an unknown amount, an exact amount of $2 sets cannot be shown; rather, a countable amount will be indicated in the diagram through special labeling. Here is a possible final diagram for the situation in Exercise. The following equation should be recorded beside the diagram: C= $ + H^$ 2h. Since H serves as the multiplier, it is written in the multiplier position, which is the first factor of the product. C $ $2 $2 H amount of $2 $2 Now consider Exercise 2 on Worksheet b: Marian took cookies to a party. She gave a third of her cookies to Adam. Adam then gave a fourth of his cookies to Charles. Charles gave half of his cookies to Barbara. If Barbara received two cookies in all, how

32 2 Math Essentials, High School Level many cookies did Marian have in the beginning? Show the initial equations. Then try to combine them into one equation that involves only the variable for Marian s amount of cookies. Since several relationships are involved in this situation, have students draw a diagram for each one. A horizontal bar should be drawn between each touching pair of diagrams. Here is a possible sequence of diagrams to use, along with their recorded equations: M A M = A 4 A C 4 A = C 2 C B 2 C = B B B = 2 Exercise 2 asks students to find a single equation that relates Marian s amount of cookies to the 2 cookies that Barbara received. Guide students to apply backward thinking to their diagrams, beginning with the last diagram. Through a substitution process, the diagrams can be stacked on each other. Looking only at the left side of each diagram and moving upward, one-half C replaces the B in the last diagram, then one-fourth A replaces C, and finally one-third M replaces A. It might be helpful for students to draw arrows on their diagrams to show where the substitutions occur. Here is a possible example, along with the combined equation that results: M A M = A 4 A C 4 A = C 2 C B 2 C = B B B = 2 2 = 2 C = 2 4 A = 2 4 M So, 2 = 2 4 M = 24 M.

33 Algebraic Thinking and Applications Answer Key for Worksheet b Suggested diagrams and their equations are provided; other formats are possible.. C= $ + H^$ 2h [The diagram is shown in the text.] 2. [See the diagrams and sequence of equations in the text.]. C = number of jelly beans in whole cup; 5 - C =28 5 C B = initial balance in bank account; B - $ 85 + $ 60 = -$ 0 B -$85 +$60 -$0 Alternate format: B $85 $60 $0 5. p = size of parking lot in square feet; p = p 5 p 5 p 200 2, d = t(r). [Note: The factor t in this case is the multiplier, so it is written first in the product; later, the commutative property might be applied to rewrite the equation in the more familiar form, d = rt.] t amount of r r r r d

34 4 Math Essentials, High School Level 7. G, L, and A: Number of tokens each person has: G= 4 + L; A= G; A= L L 2 ^ 4 + h= G L A 2 G A 2 L 8. N, N + 2: consecutive odd integers; N+ ^N+ 2h= 76 N N 76

35 5 Worksheet b Drawing Diagrams for Linear Relationships Name Date For each exercise, draw a diagram on another sheet of paper to represent each relationship in the situation. Write equations in symbols beside the diagrams.. Eddie s Dogwalking Service charges $ for each walk plus $2 per hour for each hour the dog is walked. Find an equation that shows the relationship between the number of hours walked, H, and the total cost, C, for walk. Copyright 2005 by John Wiley & Sons, Inc. 2. Marian took cookies to a party. She gave a third of her cookies to Adam. Adam then gave a fourth of his cookies to Charles. Charles gave half of his cookies to Barbara. If Barbara received two cookies in all, how many cookies did Marian have in the beginning? Show the initial equations. Then try to combine them into one equation that involves only the variable for Marian s amount of cookies.. Jaime removed one-third of a cup of jelly beans from a jar that held 5 jelly beans at first. She recounted and found that there were still 28 jelly beans in the jar. Approximately how many jelly beans would fill a whole cup? 4. On Friday, Sam wrote a check for $85. The following Monday, he deposited $60 into his bank account. On Wednesday, he checked his bank s Web site and learned that he had overdrawn his account by $0. If Sam made no other transactions between Friday and Wednesday, what was his balance before he wrote the check on Friday? 5. Three-fifths of a parking lot is scheduled to be resurfaced with new asphalt. Another 200 square feet of driveway will also be resurfaced at that time. The contractor has agreed to repave 2,900 square feet total. What is the size of the parking lot in square feet? Hint: To show a large quantity in a diagram, write the number inside a rectangle. 6. Let r represent the average speed in miles per hour that a car traveled on a trip. Let d represent the distance in miles that the car had traveled t hours after the beginning of the trip. Find an equation that relates the distance traveled to the speed and the time traveled. 7. Gary has 4 more game tokens than Leo has. Angie has half as many tokens as Gary. Find an equation that relates Angie s tokens to Leo s tokens. 8. Two consecutive odd integers have a sum of 76. What are the two integers?